Boundedness of rough integral operators on Triebel-Lizorkin spaces

Abstract

Abstract:We prove the boundedness of several classes of rough integral operators on Triebel-Lizorkin spaces. Our results represent improvements as well as natural extensions of many previously known results.2010 Mathematics Subject Classification: Primary: 42B20; Secondary: 42B15, 42B25.

A note on maximal singular integrals with rough kernels

“…During the last several years, a considerable amount of attention has been given to investigate the boundedness for various kinds of integral operators on Triebel-Lizorkin spaces. For examples, see [1,4,5] for singular integrals, [15,17,26,27] for Marcinkiewicz integrals, [16,27] for Littlewood-Paley functions, [14,18,20] for maximal functions, and [21,22] for maximal singular integrals. The main purpose of this paper is to prove the boundedness and continuity of the maximal singular integral and maximal operators related to homogeneous mappings on Triebel-Lizorkin spaces when their kernels are given by function in the Hardy space H 1 (S n-1 ).…”

“…Let S n-1 be the unit sphere in R n equipped with the induced Lebesgue mea-sure dσ . Let be integrable over S n- 1 and satisfy S n-1 (u) dσ (u) = 0.…”

A note on Marcinkiewicz integrals supported by submanifolds

“…During the last several years, a considerable amount of attention has been given to investigate the boundedness for several integral operators on the Triebel–Lizorkin spaces and Besov spaces. For examples, see [ 1 – 6 ] for singular integrals, [ 7 – 13 ] for Marcinkiewicz integrals, [ 14 ] for the Littlewood–Paley functions, [ 15 – 18 ] for maximal functions. In this paper we continue to focus on this topic.…”

“…Recently, Yabuta [ 10 ] improved and extended the above results to the case and for some . For other interesting work on this topic we refer the reader to [ 1 , 7 , 8 , 28 – 33 ].…”

Boundedness of certain singular integrals along surfaces on Triebel–Lizorkin spaces

“…For examples, Chen, Fan and Ying [9] proved that T is bounded on P F p;q .R n / provided 2 L r .S n 1 / for some r > 1; Al-Qassem, Cheng and Pan [2] (also see [10]) improved the results of [9] to the case 2 Fˇ.S n 1 /; Chen and Ding [11] showed that T is bounded on P F p;q .R n / under the assumption of that 2 H 1 .S n 1 /; see [12,20,21] for more relevant results. Indeed, lots of attentions have been paid to this topic.…”